y = \tanh x

Graph of the Hyperbolic Tangent $y = \tanh x$

The hyperbolic tangent function $y = \tanh x$ is defined as the ratio of the hyperbolic sine to the hyperbolic cosine¹.

\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}

It is the hyperbolic counterpart of the ordinary tangent $\tan x$ .

Domain and range

The domain is all real numbers
The range is the open interval $(-1, 1)$
It increases monotonically
It is an odd function

The denominator $\cosh x$ is always at least $1$ and never $0$ , so the function is defined for every real number. Where $\tan x$ had asymptotes and a range covering the whole line, $\tanh x$ is smooth and bounded.

Symmetry

Since $\tanh(-x) = -\tanh x$ , the function is odd and its graph is symmetric about the origin.

Monotonicity

The derivative is as follows.

\frac{d}{dx}\tanh x = \operatorname{sech}^2 x = 1 - \tanh^2 x

It is always positive, so the function is strictly increasing on the whole real line. The slope is steepest at the origin, where it equals $1$ , and flattens toward both ends, producing a smooth S-shaped curve.

The second derivative is $-2\tanh x\,\operatorname{sech}^2 x$ , which changes sign only at the origin. The single inflection point is $(0, 0)$ , and it is also the center of symmetry.

Asymptotes and notable values

$x$	$\tanh x$
$\to -\infty$	$\to -1$
$0$	$0$
$1$	$\approx 0.7616$
$2$	$\approx 0.9640$
$3$	$\approx 0.9951$
$\to +\infty$	$\to 1$

The lines $y = 1$ and $y = -1$ are horizontal asymptotes, and the curve never crosses them. By $x = 3$ the value already exceeds $0.995$ , so the saturation is very quick.

Relation to the sigmoid

It is tied to the logistic sigmoid $\sigma(x) = \dfrac{1}{1 + e^{-x}}$ by the following identity².

\tanh x = 2\sigma(2x) - 1

Compressing the sigmoid horizontally by a factor of $\dfrac{1}{2}$ , doubling it vertically and lowering it by $1$ gives $\tanh$ . The only difference is whether the range is $(0, 1)$ or $(-1, 1)$ ; the shape is the same.

Applications

An activation function in neural networks
A saturating function that confines values to a range
A model for magnetization or for the response of a signal
The addition of velocities in special relativity, through rapidity

Hyperbolic functions, Wikipedia
Logistic function, Wikipedia

Graph of the Hyperbolic Tangent y=tanh⁡xy = \tanh xy=tanhx